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Combinatorial thinking
The Power of Combinatorics In Learning Abstract Today I want to talk about how powerful making neural connections can be, and why I think most students these days don’t spend enough time on this process. First the definition of combinatorics grabbed from Wikipedia: Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. So what are some examples showing the powers of combinatorics? Examples of the power of combinatorics: DNA and the Latin alphabet One example is from the book What Is Life? with Mind and Matter and Autobiographical Sketches’ by Erwin Schrödinger, Roger Penrose (Foreword). The example is that of DNA, namely how such small and low amount of units can produce such big results like us humans. Answer: combinatorics. But how can we imagine this concept better? Well, let’s take a look at the Latin alphabet: A, B … Z. With just 26 letters, we can make hundreds of thousands of words, and this is not where combinatorics comes to an end. With those words, we can make millions of combinations, which makes up sentences. Sentences make up books and ideas, and so on. We can quickly see how combinatorics doesn’t only play at the level of the Latin alphabet and neither does it only operate at the level of DNA. The things quickly grow exponentially. So I hope with these examples, the answer to the question “How does such tiny units as DNA produce such big results as us humans?” becomes more obvious, namely thanks to combinatorics. Again, if we go back to the definition grabbed from Wikipedia above, combinatorics is also used in evolutionary biology. The power of combinatorics in learning: making neural circuits and neural connections You can probably see a lot on the internet how people are saying that the more knowledge you have, the faster you learn. In my experience, that’s pretty much true. The question now becomes, how can we make something that is already exponential even more exponential? Well, one thing I personally deliberately perform whenever learning things, is asking myself “What does this remind me of?” Actually, this process is almost happening subconsciously by now. One answer that this question brought me to, is this chapter. The book didn’t explicitly tell me how combinatorics enhances learning, I had to make that connection myself, and here I am, making hundreds more connections when typing. Another way to make use of combinatorics, is to simply try to make one connection between two concepts e.g. combinatorics and learning. Once you have made that connection, simply think about it, write about it, and maybe even teach others. These practices enable you to use the power of combinatorics in a more subconscious manner rather than consciously i.e. making the connection between combinatorics and learning consciously. What do I mean by “in a subconscious manner?” Well, everything that I am writing about right now is flowing impromptu. Again, the only connection I consciously made was between combinatorics and learning. All these practices creates new neural circuits and connections. There are many more ways to make use of the power of combinatorics, but I suggest reading these two chapters of mine: 10/06/2019 — My Learning Trajectory, Chapter Two: Things That Increase The Quality of My Learning and 10/12/2019 — My Learning Trajectory, Chapter Three: How I Learn This also seems to immediately give an answer to the question I had for a long time already: “How do these little practices give such big results in my life?” Again, thanks to me writing this chapter right now, the unplanned answer seems to be combinatorics. Thanks to combinatorics I am able to have all this information at such a young age. Flaw of public education: not making use of combinatorics So what could be the other extreme end, the opposite of combinatorics? I guess this quote would suffice: “The telephone book is full of facts, but it doesn’t contain a single idea.” — Mortimer Adler I honestly think public education relies too much on standardized testing and rote memorization, things that don’t require students to know how to make use of the power of combinatorics. Why would you, as a student, spend an extra 2 hours a day to make combinations between concepts when such answers and ideas won’t be present on a standardized test or bump up your grade? The school doesn’t care that you made a combination between entropy and intelligence. No, the school just wants you to learn the definition of entropy (when studying physics) or intelligence (when studying psychology), and nothing more. I really hope this will change in the future (which it probably will, look at all those YouTube videos from experts). By the way, see the chapter 10/04/2019 — Intelligential Entropy where I do make the connection between entropy and intelligence. Taking an occasional break from learning So instead of trying to learn and memorize more and more information, after a while you need to sit back, relax, and let your mind wander and daydream about everything you have soaked up. Note all your ideas down, think about them, write about them, teach them, and you will make thousands of neural connections in a short period of time that would normally take days, weeks, or even longer. This causes you to not only make use of combinatorics, but also extends your forgetting curve, the rate at which you forget information. How? By recalling the things, but also making new neural connections that will lead to the same concepts. So if one neural connection is weakened, you still have many more leading to the same concept. Lastly, it is all about sacrificing the present for a better future or “the tortoise beats the hare” concept. It doesn’t matter if you learn the things slowly right now, it will accelerate over time, exponentially. My own examples of using the power of combinatorics 08/25/2019 — Top-down thinking > bottom-up thinking and 08/31/2019 — Top-down, bottom-up thinking, sorting algorithms, and working memory → Here I made the connection how top-down thinking is more efficient in learning new things, and also why. 09/01/2019 — Possible Lack of Free Will and Psychological Therapies → Here I made the connection between a possible lack of free will and how that concept should influence psychological therapies, psychotherapists themselves, and even amongst everyday people. 09/11/2019 — Entropic Learning Model → This is a model based on the idea that if you reduce entropy locally (in your brain) by making neural connections etc., you will also reduce entropy externally (your environment). And many more examples to give, but I will keep it as it is right now. John von Neumann making use of the power of combinatorics George Dantzig is the mathematician who thought that two problems on the blackboard were homework. He solved them and handed them, albeit a bit later, so he thought they were overdue. Here’s the plot twist: They were two famous unsolved problems in statistics with which the mathematics community struggled for decades. When George Dantzig brought von Neumann an unsolved problem in linear programming “as I would to an ordinary mortal”, on which there had been no published literature, he was astonished when von Neumann said “Oh, that!” before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceived theory of duality. Afterword Remember how DNA are such tiny units yet produce such big results? Same thing with making just one connection between two concepts, doing mindfulness 20 minutes a day etc. The small things do matter. Category:Metacognition